Convex cores for actions on finite-rank median algebras (Q6621673)

A metric space \(X\) is median if, for all \(x_{1}, x_{2}, x_{3} \in X\), there exists a unique \(m=m(x_{1}, x_{2}, x_{3})\in X\) such that \(d(x_{i},x_{j}) =d(x_{i},m)+ d(m,x_{j})\) for all \(1 \leq i < j \leq 3\). The rank of a connected median space is the supremum of the topological dimensions of its compact subsets. Median spaces arise as ultralimits of sequences of CAT(0) cube complexes of uniformly bounded dimension, or any sequence of spaces that can be suitably approximated by such cube complexes. Thus, the asymptotic cones of any coarse median group can be given a natural structure of finite-rank median space.\N\NThe paper under review aims first to demonstrate the following two results:\N\NCorollary A: Let \(X\) be a connected, finite-rank median space. For every \(g \in \operatorname{Isom} X\): (1) the translation length of \(g\) is realized by some point of \(X\); (2) if \(X\) is geodesic and \(g\) does not fix a point, then \(g\) admits an axis: a bi-infinite \(\langle g \rangle\)-invariant geodesic along which \(g\) translates non-trivially.\N\NCorollary B: Let \(A\) be a finitely generated, virtually abelian group. Let \(A \curvearrowright X\) be an isometric action on a connected, finite-rank median space. Then \(A\) stabilizes a non-empty convex subset \(C \subseteq X\) isometric to a subset of \((\mathbb{R}^{n},d_{\ell^{1}})\) with \(n \leq \operatorname{rk} X\).\N\NThe reviewer points out the somewhat surprising fact that these results do not require the metric on \(X\) to be complete, moreover most of the results in this paper hold for general automorphisms of finite-rank median algebras, even when no metric or topology is present.\N\NThe author then shows that every action of a finitely generated group on a finite-rank median algebra admits a non-empty ``convex core'' (see Definition 3.15). He uses these results to deduce an analogue of the flat torus theorem for actions on connected finite-rank median spaces. Furthermore, he proves that isometries of connected finite-rank median spaces are either elliptic or loxodromic.